Lewis on common knowledge

The reading for today is chapter II, section 1 of Convention.

In it, Lewis discusses a state of affairs, A, “you and I have met, we have been talking together, you must leave before our business is done; so you say you will return to the same place tomorrow.” Lewis notes that this generates expectations and higher order expectations: “I expect you to return. You will expect me to expect you to return. I will expect you to expect me to expect you to return. Perhaps there will be one or two orders more”. His task is to explain how these expectations are generated.

We’ll just be looking at the first few steps of his famous proposal, which are framed in terms of reasons to believe. It has three premises:

1. You and I have reason to believe that A holds.
2. A indicates to both of us that you and I have reason to believe that A holds.
3. A indicates to both of us that you will return.

“Indication” is defined counterfactually: A indicates to someone x that Z iff if x had reason to believe that A held, x would thereby have reason to believe that Z. Lewis notes that indication depends on “background information and inductive standards” of the agent in question. The appeal to inductive standards might suggest a somewhat subjective take on epistemic reasons is in play here, but even if you think epistemic reasons are pretty objective, the presence or absence of a belief in defeaters to inductive generalizations, for example, will matter to whether that counterfactuals of this form are true.

(I’m not sure about the significance of the “thereby” in this statement. Maybe Lewis is saying that the reason for believing that A held would also be the reason for believing that Z is the case. I’m also not sure whether or not this matters).

There follows a passage that I have difficulty following. Here it is in full.

“Consider that if A indicates something to x, and if y shares x’s inductive standard and background information, then A must indicate the same thing to y. Therefore, if A indicates to x that y has reason to believe that A holds, and if A indicates to x that Z, and if x has reason to believe that y shares x’s information, then A indicates to x that y has reason to believe that Z (this reason being y’s reason to believe that A holds)”.

In this passage, we first get the following inference pattern (I):

1. A indicates p to x.
2. x and y share standard/background information.
3. Conclusion: A indicates p to y.

That seems fair enough.

Following the “therefore”, we get the following inference (II):

1. A indicates that [y has reason to believe that A holds] to x.
2. A indicates that Z to x.
3. x has reason to believe that x and y share standards/background information.
4. Conclusion: A indicates that [y has reason to believe that Z] to x.

This is a complex piece of reasoning, and it’s relation to the earlier inference pattern is not at all clear. For example, in the first inference pattern, facts about shared standards are mentioned. In the second, what we have to work with is x having reason to believe that there are shared standards. This prevents us directly applying argument I to derive argument II. Some work needs to be done to connect these two.

Given the validity of the first pattern you can plausibly argue for the goodstanding of the following derived pattern (III):

1. x hass reason to believe that A indicates p to x.
2. x has reason to believe that x and y share standard/background information.
3. Conclusion: x has reason to believe that A indicates p to y.

Now III.2 is also II.3, so we now hope to connect the arguments. But the other two premises are not facts about what x has reason to believe, as they would have to be in order to apply III directly. Rather, they are facts about what A indicates to x.

We need to start attributing enthymetic premises. Perhaps there is a transparency assumption, namely that IV is valid:

1. A indicates p to x.
2. x has reason to believe that A indicates p to x.

IV allows us to get from II.2 to the claim that x has reason to believe that A indicates Z to x. And you can then use II.3 to supply the remaining premise of inference pattern III. What we get is the following: x has reason believe that A indicates Z to y. And so we could argue that argument II was a good one, if the following inference pattern was good  (V):

1. A indicates that [y has reason to believe that A holds] to x.
2. x has reason to believe that [A indicates Z to y].
3. Conclusion: A indicates that [y has reason to believe that Z] to x.

The conclusion of V is the same as that of II. V.1 is simply II.1, and we have seen that III and IV get us from II.2 and II.3 to V.2. So the validity of V suffices for the validity of II. So what do we think about inference pattern V?

V is, in fact, an inference pattern that Cubitt and Sugden, in their very nice analysis of Lewis’s argument, take as one of the basic assumptions (they give it as a material conditional, and label it A6). It seems really dubious to me however.

The reason that it looks superficially promising is because the three embedded claims  constitute a valid argument, and the embedding contexts looks like we’re reporting the validity of this argument “from x’s perspective”. The embedded argument is simply the following: If y has reason to believe that A holds, and A indicates Z to y, then y will have reason to believe that Z holds. Given the way Lewis defined indication in terms of the counterfactual condition, this is just a modus ponens inference.

Now this would be exactly the right diagnosis if we were working not with V but with VI:

1. A indicates that [y has reason to believe that A holds] to x.
2. A indicates that [A indicates Z to y] to x.
3. Conclusion: A indicates that [y has reason to believe that Z] to x.

VI really does look good, because each premise tells us that the respective embedded clause is true in all the closest worlds where x has reason to believe that A holds. And since the final embedded clause follows logically from the first two, it must hold in all the closest worlds where x has reason to believe A. And that’s what the conclusion of VI tells us is the case.

But this is irrelevant to V. V.2 doesn’t concern what x has reason to believe some counterfactual worlds, but what they have reason to believe in the actual world. And for all we are told, in the closest worlds where x has reason to believe that A is the case, they may not have reason to believe some of the things they actually have reason to believe. That is: A might be the sort of thing that defeats x’s reason to believe that A indicates Z to y. So this way of explaining what’s going along fails.

So I’m not sure how best to think about Lewis’s move here. The transition he endorses between I and II really isn’t transparently good. A natural line of thought leads us to think of him resting on Cubitt and Sugden’s reconstructed premise A6, V above. But that really doesn’t look like something we should be relying on.

Is there some other way to understand what he has in mind here?